2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160976For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost sure central limit theorem (ASCLT). In fact, we provide a speed of convergence in the Kantorovich metric. Maxima of partial sums are also shown to obey an ASCLT. The key-tool is an exponential inequality recently obtained. Then we derive almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdos-Renyi type law). We end up with an application to entropy estimation ASCLT's that refi ne Shannon-McMillan-Breiman and Ornstein-Weiss theorems.25p; title has changed; minor corrections; published in Erg. Th. Dynam. Sys. (2005)ProbabilityDynamical Systemstext